cp's OEIS Frontend

Strings that can be reduced to null string by repeatedly removing an entire run of two or more consecutive symbols.

Here we are using Lenormand's "raboter" map in a stricter sense than in A318921 and A319419. If S is a binary string with successive runs of lengths b,c,d,e,..., the "raboter" map sends S to the binary string with successive runs of lengths b-1,c-1,d-1,e-1,... Runs of length 0 are omitted (they are indicated by dots in the examples below).

To get a(n), start with S equal to the binary expansion of n beginning with the most significant bit, and keep applying the map until we reach the empty string.

After the first step, the string may start with a string of 0's: this is acceptable because we are working with strings, not binary expansions of numbers.

For example, 34 = 100010 -> .00.. = 00 -> 0. = 0 -> . (the empty string), taking 3 steps, so a(34) = 3.

Note: this is not the same as the number of applications of the map k -> A318921(k) needed to reduce the binary expansion of n to zero (because A318921 does not distinguish between 0 and the empty string).

This is also not the same as the number of applications of the map k -> A319419(k) needed to reduce the binary expansion of n to -1 (because A319419 does not distinguish between a string of 0's and a single 0).

The value k appears for the first time when n = 2^k - 1.

Digits that appear more than once in n are erased. Leading zeros are erased unless the result is 0. If all digits are erased, we write 0 for the result (A320485 is another version, which uses -1 for the empty string).

More than the usual number of terms are shown in order to reach some interesting examples.

a(n) = 0 mostly. - David A. Corneth, Oct 24 2018

The number of d-digit numbers n for which a(n) > 0 is at most d*9^d, so in this sense most a(n) are 0. - Robert Israel, Oct 24 2018

The set of numbers with the property that their digits appear at least twice is of asymptotic density 1 (and the set of numbers that have a digit that occurs only once is of density 0), so in that sense it is rather exceptional for large n to have a(n) > 0. - M. F. Hasler, Oct 24 2018

a(n)=1 plus the number of symbol changes in the first n terms of A000002. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010

From N. J. A. Sloane, Nov 12 2018: (Start)

This seems to be A001462 rewritten so the run lengths are given by A000002. The companion sequence, A000002 rewritten so the run lengths are given by A001462, is A321020.

Note that Kolakoski's sequence A000002 and Golomb's sequence A001462 have very similar definitions, although the asymptotic behavior of A001462 is well-understood, while that of A000002 is a mystery. The asymptotic behavior of the two hybrids A156253 and A321020 might be worth investigating. (End)

To expand upon N. J. A. Sloane's comments, it's worth noting that Golomb's sequence has a formula from Colin Mallows: g(n) = g(n-g(g(n-1))) + 1, which closely resembles a(n) = a(n-gcd(a(a(n-1)),2)) + 1. - Jon Maiga, May 16 2023

Cuts-resistance is defined in A319416.

This triangle summarizes the data shown in A319420.

Conjecture (Sloane): Sum_{i = 1..n} i * T(n,i) = A189391(n).

Digits that appear more than once in n are erased. Leading zeros are erased unless the result is 0. If all digits are erased, we write -1 for the result.

The map n -> a(n) was invented by Eric Angelini and described in a posting to the Sequence Fans Mailing List on Oct 24 2018.

More than the usual number of terms are shown in order to reach some interesting examples.

a(n) = -1 mostly. - David A. Corneth, Oct 24 2018

The cells are the squares of the standard square grid.

Cells are either OFF or ON, once they are ON they stay ON forever.

Each cell has 8 neighbors, the cells that are a knight's move away.

We begin in generation 1 with a single ON cell.

A cell is turned ON at generation n+1 if it has exactly one ON neighbor at generation n.

(Since cells stay ON, an equivalent definition is that a cell is turned ON at generation n+1 if it has exactly one neighbor that has been turned ON at some earlier generation. - N. J. A. Sloane, Dec 19 2018)

This sequence has similarities with A151725: here we use knight moves, there we use king moves.

This is a knight's-move version of the Ulam-Warburton cellular automaton (see A147562). - N. J. A. Sloane, Dec 21 2018

The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections, which generate the dihedral group D_8 of order 8), so A319019 is a multiple of 8 (compare A322050). - N. J. A. Sloane, Dec 16 2018

From Omar E. Pol, Dec 16 2018: (Start)

For n >> 1 (for example: n = 257) the structure of this sequence is similar to the structure of both A194270 and of A220500, the D-toothpick cellular automata of the second kind and of the third kind respectively. The animations of both CAs are in the Applegate's movie version.

Also, the graph of A319018 is a bit similar to the graph of A245540, which is essentially a 45-degree-3D-wedge of A245542 (a pyramid) which is the partial sums of A160239 (Fredkin's replicator). See "Plot 2": A319018 vs. A245540. (End)

The conjecture that A322050(2^k+1)=1 also suggests a fractal geometry. Let P_k be the associated set of eight points. It appears that P_k may be written as the intersection of four fixed lines, y = +-2*x and x = +-2*y, with a circle, x^2 + y^2 = 5*4^k (see linked image "Log-Periodic Coloring"). - Bradley Klee, Dec 16 2018

In many of these toothpick or cellular automata sequences it is common to see graphs which look like some version of the famous blancmange curve (also known as the Takagi curve). I expect that is what we are seeing when we look at the graph of A322049, although we probably need to go a lot further out before the true shape becomes apparent. - N. J. A. Sloane, Dec 17 2018

The graph of A322049 (related to first differences of this sequence) appears to have rather a self-similar structure which repeats at powers of 2, and more specifically at 2^10 = 1024. There is no central symmetry or continuity, which are characteristic properties of the blancmange curve. - M. F. Hasler, Dec 28 2018

The 8 points added in generation n = 2^k + 1 are P_k = 2^k*K where K = {(+-2, +-1), (+-1, +-2)} is the set of the initial 8 knight moves. So P_k is indeed the intersection of the rays of slope +-1/2 resp. +-2 and a circle of radius 2^k*sqrt(5). In the subsequent generation n = 2^k + 2, the new cells switched on are exactly the 7 "new" knight move neighbors of these 8 cells, (P_k + K) \ (2^k - 1)*K. The 8th neighbor, lying one knight move closer to the origin, has been switched on in generation 2^k, together with an octagonal "wall" consisting of every other cell on horizontal and vertical segments between these points (2^k - 1)*K, and all cells on the diagonal segments between these points, as well as 2 more diagonals just next to these (on the inner side) and shorter by 2 cells (so they are empty for k = 1). This yields 4*(2 + (2^k - 2)*(1+3)) new ON cells in generation 2^k, plus 8*(2^(k-1) - 2) more new ON cells on horizontal, vertical and diagonal lines 4 units closer to the origin for k > 2, and similar additional terms for k > 4 etc. - M. F. Hasler, Dec 28 2018

This is equally the minimal number of steps to reach n from 1 using "Choix de Bruxelles", version 1 (cf. A323286), or -1 if n cannot be reached.

n cannot be reached if its final digit is 0 or 5, but all other numbers can be reached (see comments in A323286).

Sequence gives the number of edge-to-edge regular-polygon tilings having n vertex classes relative to the symmetry of the tiling. Allows tilings with two or more vertex classes having the same arrangement of surrounding polygons (vertex type), as long as those classes are distinct within the symmetry of the tiling .

There are eleven 1-uniform tilings (also called the "Archimedean" tessellations) which comprise the three regular tessellations (all triangles, squares, or hexagons) plus the eight semiregular tessellations.

Number of representations of n as the difference of two distinct triangular numbers, plus any multiple of the order of the larger triangular number.

From Jeremy Lovejoy, Nov 10 2022: (Start)

For n > 0, a(n) is also equal to the Hurwitz class number H(8n-1).

a(n) is also equal to the number of partitions y of n having no repeated even parts and smallest part odd, counted according to the weight w(y) = (-1)^(the number of even parts)*(the number of occurrences of the smallest part). For example, the partitions of 6 having no repeated even parts and smallest part odd are [5,1], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,1,1,1,1], and [1,1,1,1,1,1], which are counted with weights 1,-2,2,-1,3,-4, and 6, giving a(6) = 1-2+2-1+3-4+6 = 5. (End)